On countably compact free abelian groups: Infinite products and non-homeomorphic group topologies
by
Artur Hideyuki Tomita
Universidade de São Paulo

Is every countably compact cancellative topological semigroup a topological group? This was asked by Wallace in 1953. The first counterexample was obtained in 1994 by Robbie-Svetlichny under CH.

We showed the existence of a countably compact free abelian group without non-trivial convergent sequences under MA_{\sigma-centered}). A corollary of this result is the existence of a counterexample for the Wallace Problem under MA_{\sigma-centered}).

We show that no infinite free abelian group has the \omega-th power countably compact. In particular, there are neither infinite sequentially compact free abelian groups nor infinite p-compact free abelian groups.

We also show that an infinite subsemigroup of a free abelian group without non-trivial convergent sequences or without the identity has the \omega-th power not countably compact. In particular, neither our counterexample for the Wallace Problem mentioned above nor Robbie-Svetlichny's has the \omega-th power countably compact.

Using different methods, we prove that the Wallace Problem has a counterexample under MA_countable. We show that this counterexample cannot have the c-th power countably compact. More generally, we showed that every p-compact cancellative semigroup is a group, therefore there is no Tychonoff p-compact counterexample for the Wallace Problem. This answer a question from D. Grant.

This work has been partially supported by CNPq and University of São Paulo - São Paulo, Brazil

Date received: February 9, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caak-30.